1

Loaded microwave cavity for compact vapor-cell clocks

Michele Gozzelino, Salvatore Micalizio, Claudio E. Calosso, Aldo Godone, Haixiao Lin and Filippo Levi

Vapor-cell devices based on microwave interrogation providea stable frequency reference with a compact and robust setup.Further miniaturization must focus on optimizing the physicspackage, containing the microwave cavity and atomic reservoir.In this paper we present a compact cavity-cell assembly basedon a dielectric-loaded cylindrical resonator. The structure ac-commodates a clock cell with 0.9 cm3 inner volume and hasan outer volume of only 35 cm3. The proposed design aims atstrongly reducing the core of the atomic clock, maintaining atthe same time high-performing short-term stability (σy(τ) ≤5×10−13 τ−1/2 standard Allan deviation). The proposed struc-ture is characterized in terms of magnetic field uniformity andatom-field coupling with the aid of finite-elements calculations.The thermal sensitivity is also analyzed and experimentallycharacterized. We present preliminary spectroscopy results byintegrating the compact cavity within a rubidium clock setupbased on the pulsed optically pumping technique. The obtainedclock signals are compatible with the targeted performances.The loaded-cavity approach is thus a viable design option forminiaturized microwave clocks.

I. INTRODUCTIONVapor-cell atomic clocks provide excellent frequency stabil-

ity together with low volume and power consumption. Theyare employed both on ground and in space applications [1].Laser-based devices, working either with continuous or pulsedschemes, have shown state-of-the-art stability levels in theirclass, making them attractive options for next generationglobal navigation systems [2–4]. Several industries have shownincreasing interest for the pulsed optically pumped (POP) Rbclock, in virtue of its demonstrated performances and maturetechnology [5].

For industrial and spaceborne applications, a compact andlight physics package is of high importance, not only forweight and volume considerations, but also for the possi-bility to reduce the overall power consumption and to im-prove the mechanical design. Moreover, for space applica-tions, the mid and long-term stability is of greater rele-vance than having state-of-the-art short term performances('1.5× 10−13 τ−1/2) [6,7]. The development of a compactphysics package goes in this direction, providing easier tem-perature stabilization and mitigation of temperature gradients,compared to a distributed object. This results in foreseenimproved mid-term frequency stability performances [8].

This work presents a miniaturized microwave cavity suitablefor POP [9] or continuous-wave [7] Rb clocks, looking for a

M. Gozzelino, S. Micalizio, C. E. Calosso, A. Godone, H. Lin, and F. Leviare with Istituto Nazionale di Ricerca Metrologica, INRIM, Torino, Italy.E-mail: s.micalizio@inrim.it

H. Lin is with Key Laboratory of Quantum Optics, Shanghai Institute ofOptics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800and with University of Chinese Academy of Sciences, Beijing 100049, China.Lin H. acknowledges the China Scholarship Council (CSC) and the NationalNatural Science Foundation of China (NSFC) under Grant No. 91536220.

trade-off between physical dimensions and desired short-termstability performances.

The physics package of Rubidium standards typicallypresents a layered structure, including several layers of mag-netic and thermal shields. To reach an overall reduction thepackage size, the most efficient strategy is to reduce thevolume of the physical core of the clock, i.e. the microwavecavity. Many high-performing clocks make use of a cylindricalcavity resonant on the TE011 mode, because of its favorableH-field distribution, which is uniform and parallel to the cavityaxis in the central region of the cavity. This features guaranteesto excite the clock transition in an efficient way, increasing thecontrast of the clock signal [10]. The cavity inner dimensionsare designed in order to tune the resonance frequency to theatomic clock transition (6.8 GHz for 87Rb).

One way to shrink the cavity dimensions, keeping themagnetic field resonant with the atomic clock transition, isthe use of a loop-gap resonator (also called “magnetron” or“split-ring” resonator) [11,12]. This approach has been proveneffective both for the continuous-wave and the POP rubidiumfrequency standards [13–15]. Indeed, it provided reduction ofthe overall cavity-cell volume up to a factor 4.5, retainingcomparable cell size and clock performances, compared to thetraditional cylindrical cavity.

Inserting a dielectric material inside the cavity volume canalso lead to remarkable volume reduction [16,17]. Dielectricloading can also be exploited, to some extent, to increase thefield uniformity in the active volume [18,19].

We propose a novel design solution for the cavity-cellassembly based on an alumina-loaded microwave cavity,demonstrating an external volume of only 35 cm3. The loadedcavity still works on a TE011-like mode, ensuring favorablemagnetic field uniformity and directionality. Such strong sizereduction is possible by scaling also the clock cell, whoseinner volume is reduced by a factor 8. As demonstrated inmore detail in Section II and Section III, this volume reductionpartially affects the clock short-term stability, but still providesinteresting performances.

The proposed alumina-loaded cavity offers new designalternatives and thus facilitates the use of the POP technologyfor in-field applications. Advantages of this approach is the no-table size reduction and mechanical stability, as the dielectriccan serve also as self-centering spacer for the clock cell.

The paper is organized as follows: in Section II the pro-posed design is introduced. In Section II-A the loaded-cavitymain features are analyzed with the aid of Finite ElementMethod (FEM) analysis. Finally, in Section III the cavity isexperimentally characterized, and foreseen short-term stabilityperformances for such a cavity in a POP clock experiment arediscussed.

arX

iv:2

005.

1015

9v1

[ph

ysic

s.at

om-p

h] 2

0 M

ay 2

020

2

0.047" coax SMPM connector

below cut-offwaveguide

Aluminum cavitybody Alumina

tube

Teflon spacers

38 mm

32 mmfocusing lens

Quartz cell

photodiode

Figure 1: Rendering of the cavity-cell assembly (cross-sectionview).

II. CYLINDRICAL LOADED-CAVITY DESIGN

Reducing the cell dimensions has two effects: first, itreduces the available atomic sample volume, thus decreasingatom number and the signal-to-noise ratio. Second, the atomicpopulation relaxation rates are increased, due to the collisionswith the cell walls. The first point is not critical for many laser-based vapor-cell clocks, since they are not limited by shot-noise, rather from laser intensity and frequency noises [20].The second issue can be mitigated by finding an optimal buffergas pressure for which the total relaxation rate is minimized.Indeed, since for typical operational temperatures the maintransverse relaxation-rate contribution is due to spin-exchangecollision between Rb atoms, there is margin to increase thepressure of the buffer gas without significantly enhancing thetotal relaxation rate [21].

Given the last considerations, for a traditional buffer-gasmixture composed of Ar and N2 (1.6:1 ratio) [22], the totalbuffer-gas pressure is set to 40 torr (53.2 hPa). At this pres-sure, and for a typical operational temperature of 62 ◦C, thetransverse relaxation rate γ2 is minimized (γ2 ' 400 s−1) [23].Compared to the case of a 2 cm× 2 cm cylindrical cell with25 torr of buffer gas (γ2 = 280 s−1), this corresponds to a1.4 factor increase in the transverse relaxation rate. The γ2rate causes a decay of the atomic signal, ultimately limitingthe interrogation time length (T ≈ γ−12 ), so we expect theoptimal Ramsey time to be reduced roughly by the same factor(T ' 2.4 ms versus T ' 3.4 ms). Finally, a reduction ofthe interrogation time lowers the atomic line quality factor,impacting the short-term stability. We consider these valuesa reasonable trade-off between expected performance andphysics package size reduction.

The complete setup is shown in Fig. 1. The dielectrichas a cylindrical shape, to preserve as much as possible thesymmetry of the system. A small indent is introduced to allowthe cell stem to exit the cavity volume (a cold point is desirableto induce metallic rubidium condensation outside the cavityvolume). The alumina tube is centered with the aid of twoTeflon rings that increase the mechanical tolerance. The cavityis made of aluminum and consist of a cylindrical body andtwo endcaps fixed by a set of screws. The end-caps of thecavity present two circular apertures, with 9 mm diameter,

Table I: Resonance frequencies and Q-factor for the maincavity modes (without clock cell).

mode ν0 (FEA)1 Qi (FEA) ν0 (exp.) Qi (exp.)

TE011 7.17 GHz 5200 7.00(1) GHz 4000 ± 200TM111 7.58 GHz 5100 7.50(1) GHz 3200 ± 200TM011 6.20 GHz 4200 – –

for optical access. These apertures are followed by cylindricalwave-guides with the same diameter. With such a geometry,the TE011 mode frequency is below the waveguide cut-offfrequency and power attenuation of at least 40 dB for theevanescent microwave field is obtained with a total waveguidelength of 12 mm, making the clock frequency instability due tomicrowave leakage negligible. Also in this case, the reductionof the cell size and, consequently, of the needed optical accessallowed us to scale the waveguides volume, maintaining thesame attenuation of the unloaded design. On one side of thecavity, a magnetic coupling loop (created with a short-circuitedSMPM coaxial cable) provides the microwave excitation. Thesystem is completed with a plano-convex lens which focusesthe laser light onto a 2.6 mm× 2.6 mm Si photodiode.

A. Finite element analysis (FEA)

The resonance frequency and spatial distribution of thecavity modes of interest are analyzed with the aid of the finite-element-method software tool CST-studio [24].

At first, the geometry of the cavity, cell and loading materialare determined, so that the TE011-mode frequency matches theatomic frequency. A check of the nearest resonance modes ismade, to verify that they do not overlap with the main cavitymode.

From the FEA analysis, given their spatial distribution, thetwo nearest modes are recognized as the TM011 and TM111.They lie at least 400 MHz away from the eigenmode of interestfor the clock operation. Conveniently, the dielectric loadingcompletely lifts the degeneracy of the TM111 and TE011modes, thus no mode-choke for the TM111 mode is necessary.In Table I a list of the simulated resonance frequency modesand relative intrinsic Q-factors (considering also the dielectriclosses from the loading materials) is shown for the loadedmicrowave cavity without the clock cell. The simulated valuesare compared to the measured one. As it will also be noted inSection III, the discrepancy in the absolute frequency is mainlydue to the value of the dielectric constant of the alumina at6.8 GHz, which resulted 3 % higher than the value given fromthe manufacturer [25].

Second, the uniformity and directionality grades of theTE011 mode is evaluated. In particular a “uniformity coeffi-cient”, introduced in [10], is used to characterize the amplitudevariations of the Hz component over the active volume (Va).To avoid ambiguity, the latter is taken coincident with thecell inner volume. Compared to [10], the definition has beenslightly modified to allow the maximum of the magnetic field

3

(a) H field in the y-z plane.

(b) Ex component in the y-z plane.

Figure 2: Magnetic and electric field distribution inside thecavity volume for the TE011 mode.

to lie away from the center of the cavity (due to asymmetriesin the cell and loading materials):

u =1Va

∫VaHz(r)

2 dV

maxr∈Va{Hz(r)2}

(1)

where r is the spatial coordinates vector. In this way0 < u < 1, with u = 1 corresponding to a perfectly uniformfield distribution. Following [13], we also report the “orienta-tion coefficient” ξ, defined as:

ξ =

∫VaHz(r)

2 dV∫Va|H(r)|2 dV

(2)

This coefficient is a figure of merit of the orientation of the Hfield along the quantization axis z. A high orientation factor(ξ close to 1), minimizes the excitation of the σ-transitions(∆F = 1, ∆mF = 1), that can cause unwanted cavitypulling on the clock transition [26]. For completeness, we alsoreport the filling factor, which is mostly important for activeoscillators (or POP with microwave detection [27]), expressingthe degree of coupling between the microwave field and theatomic sample:

η′ =

(∫VaHz(r) dV

)2Va∫Vc|H(r)|2 dV

(3)

where Vc is the inner cavity volume. In Fig. 2a, the H fieldlines and amplitude for the proposed loaded-cavity assemblyare shown in the y-z plane (the pumping/detection laserpropagates along the z-axis). We can notice a rather gooduniformity of the field component over the cell volume. Wepoint out that along this plane we have the maximum fielddistortion due to the presence of the stem, while on the x-z

Table II: Comparison of different published cavity-cell assem-blies in terms of magnetic field uniformity and orientation.

cavity type u ξ η′

cylindrical [9,10] 0.59 0.92 0.38

magnetron [13] n/a 0.87 0.14

magnetron [15] n/a 0.90 n/a

Al2O3-loaded (this work) 0.82 0.76 0.20

plane we have better uniformity. Looking at the field linesdistribution, we notice that a gap between the alumina tube andthe cavity wall is not only preferable in terms of mechanicaltolerances, but it also increases the mode uniformity. Indeed,due to the high dielectric constant of Al2O3, the electric fieldis strongly concentrated inside the dielectric volume [28] (seeFig. 2b) and the magnetic field lines need a dielectric-freevolume to concatenate the electric field lines.

In Table II the previously introduced coefficients forthe proposed configuration are reported and compared toother cavity-cell assemblies present in the literature. We canobserve that the uniformity is increased compared to the caseof the traditional cylindrical cavity while the orientation iscomparable to other kind of compact assemblies. Finally,the filling factor is reduced compared to the case of theunloaded cavity, reducing cavity-induced sensitivities suchas cavity-pulling, but still high enough to provide sufficientcoupling between the field and the atomic sample to achieveefficient clock interrogation.

Thermal sensitivityFor clock applications, the stability of the cavity mode

frequency is of paramount importance, as it can impact theclock frequency through cavity-pulling [21]. One of the mainparameters of influence is temperature [10,19], which canchange the resonance frequency νc by thermal expansion andby affecting the dielectric properties of the materials.

The total contribution to the sensitivity of the cavity res-onance frequency to a temperature variation ∆T , for smallvariations from the operational setpoints, can be expressed asa sum of terms:

1

νc

∆νc∆T

=∑k

xkνc

∆νc∆xk

αk +∑i

�iνc

∆νc∆�i

βi (4)

where xk are the geometric dimensions of the cavity andloading materials and αk = ∆xkxk∆T the corresponding linearthermal expansion coefficient; �i is the dielectric constant ofthe i-th material inside the cavity and βi = ∆�i�i∆T the relatedthermo-dielectric coefficient.

In Table III the major contributions to the cavity frequencythermal sensitivity are expressed in relative terms. The sensi-tivity coefficients are taken from FEM calculations, that canprovide a numerical evaluation of the mode frequency as afunction of the various parameters. In Fig. 3, for example, theTE011-mode frequency is plotted as a function of the cavitylength and radius.

4

−4 −2 0 2 4(x− x0)/mm

6.6

6.8

7.0

7.2

7.4ν c

/GHz x ≡ a

x0 = 11.5 mmx ≡ d

x0 = 24 mm

νc vs length dνc vs radius a

Figure 3: TE011 resonance frequency as a function of cavitylength and radius, as derived from the finite element analysis.The red dashed line corresponds to the atomic frequency(6.8347 GHz).

Close to the working points, we find that the main con-tribution from variations of the geometric parameters is dueto the cavity radius: 1νc

∆νc∆T = −23.9 ppm/K. The sensitivity

to the cavity thermal expansion is -1.17, close to the caseof the empty cylindrical cavity cavity (equal to −1). Giventhe low thermal-expansion coefficient of alumina (αAl2O3 =6.9 ppm/K) [25]), the geometric contribution from the loadingmaterial is instead small (' −2 ppm/K).

The other major contribution to the temperature sensitivitycomes from the variation of the dielectric constant of thealumina. From the manufacturer’s datasheet [25] we extract athermo-dielectric coefficient βAl2O3 = 92 ppm/K, while fromthe FEM analysis we obtain a linear dependence of the cavity

Table III: Major contributions to the cavity thermal sensi-tivity expressed in relative terms. The geometric and thermo-dielectric contributions are listed in the upper and lower partof the table respectively, in decreasing order of importance.

xk contributionxkνc

∆νc∆xk

αk1νc

∆νc∆T

(ppm/K) (ppm/K)

cavity radius -1.04 23 -23.9

cavity length -0.13 23 -3.0

Al2O3 thickness -0.22 6.9 -1.5

Al2O3 length -0.07 6.9 -0.5

�i contribution�iνc

∆νc∆�i

βi

(ppm/K)

Al2O3 -0.43 92 -39.7

fused silica -0.03 10 -0.3

Total sensitivity -69.0

Figure 4: Loaded cavity components prior to the assembly. Inthe picture it is also shown the first cylindrical thermal shield.As a dimensional reference, a 10 euro cent coin is shown.

frequency on �, with slope �νc∆νc∆� = −0.43. This value is

close to the case of a fully loaded cavity cavity (-0.5). This isexpected, since the electric field is mostly concentrated in thedielectric volume. The contribution from the linear expansionof the fused-silica cell and effects related to the Teflon spacersare instead negligible (< 0.1 ppm/K).

Summing up all terms, the cavity resonance sensitivity isexpected to be: ∆νc/∆T = −472 kHz/K. The total sensitivityto temperature is higher than the one of the unloaded cavityby a factor 3, but considering the lower filling factor (andfor a typical cavity detuning of 500 kHz and loaded qualityfactor QL = 2800) we estimate an expected clock fractionalfrequency sensitivity to temperature of ' 5× 10−12/K fromcavity-pulling effect [21]. A temperature control at the levelof 0.5 mK is thus sufficient to reach state-of-the-art stabilityperformance. We remind that given the small size of the cavity,such a level of temperature stabilization is not hard to achieve.

III. EXPERIMENTAL CHARACTERIZATION

The cavity-cell components are shown in Fig. 4 during theassembly phase. The cell used in the experimental validationis made of fused silica, with internal diameter and length bothequal to 1 cm.

The loaded cavity resonance frequency has been experi-mentally measured by looking at the reflected power whilesweeping the microwave frequency. The loaded cavity absolutefrequency at the operational temperature of 65 ◦C is found tobe 120 MHz lower than the simulated value. This is compatiblewith a dielectric constant for the alumina material closer to9.7, rather than the nominal value (� = 9.4) which is providedby the manufacturer at 8.5 GHz [25]. A fine tuning has beenachieved by adjusting the cavity length by fractions of 1 mm.A measurement of the cavity resonance frequency at differenttemperatures lead to an experimental temperature coefficientof −473 kHz/K for the untuned cavity and of −461 kHz/Kfor the tuned cavity, in good agreement with the simulations.No significant degradation of the cavity intrinsic quality factorhas been observed in the range 25 ◦C to 70 ◦C, due to excessdeposition of metallic Rb. Indeed, Qi has a value of 3600at room temperature, decreasing to 3200 around 70 ◦C. In

5

−30 −20 −10 0 10 20 30(νc − 6.82 GHz)/MHz

0.4

0.6

0.8

1.0re

flect

edpo

wer/

a.u.

22.4 ◦C75.5 ◦C

Figure 5: TE011 resonance frequency as a function of mi-crowave detuning at two limit temperatures (ambient tempera-ture and 75.5 ◦C). The plot shows the reflected power from thecavity, detected with a microwave circulator and photodiode.

Fig. 5 the loaded-cavity resonance mode is shown for two limittemperatures. The measurements shown refer to the cavityresonance previous to the fine tuning. From this measurementwe can also determine the cavity coupling parameter β [27],which has a value of 0.2.

The proposed cavity-cell assembly performance has beentested with a POP clock scheme. The cavity has been operatedat ambient pressure, integrated into an existing structure (sameas in [29]) composed of 2 layers of thermal shielding and 3layers of magnetic shielding. A static quantization magneticfield lifts the Zeeman degeneracy and isolates the clocktransition. The optics package is the one described in [9],including a distribuited feedback laser (DFB) working on theD2 line, frequency stabilized with an external reference cellthrough saturated absorption spectroscopy. The laser frequencyis tuned to the minimum of the clock cell absorption profilefor the |F = 2〉 atomic ground-state. The pulsing is providedby an amplitude-modulated acousto-optic modulator operatingin single-pass configuration. The system is completed witha low-noise digital control and acquisition system [30] andthe microwave synthesis chain presented in [31]. In Fig. 6 ascan of the Ramsey fringes, obtained with a laser absorptionmeasurement as the microwave frequency is swept, is shown.The total cycle time is 3.35 ms, including pumping, clockinterrogation and detection. The free-evolution time T = 2 ms,while the Rabi pulses are 0.4 ms long. The laser beam is colli-mated with a gaussian waist 2w = 6 mm. The pumping poweris 4 mW, for a pulse duration of 0.4 ms. The detection pulsepower and duration are 200 µW and 0.15 ms respectively. Withsuch timings and parameters, we obtained a fringe contrast ofabout 20 %.

To characterize the atomic properties of the Rb sample in theclock cell, we measured the longitudinal relaxation time T1,directly accessible with an optical detection of the populationevolution, with the Franzen method [32,33]. By this meanswe get T1 = 1.7(2) ms at 64.5 ◦C. The transverse relaxationtime T2 is inferred by comparing envelope of the experimental

−1000 0 1000Frequency offset /Hz

0.80

0.85

0.90

0.95

1.00

abso

rptio

nsig

nal/

a.u.

Figure 6: Ramsey scan of the atomic line (absorption signalas a function of the microwave detuning from the atomicfrequency; each data point is the result of 3 averages). Cavitytemperature 64.5 ◦C, free-evolution time T = 2.0 ms, Rabipulses length t1 = 0.4 ms.

Ramsey fringes to the one computed with the theory developedin [34]. From this analysis, it turns out that T2 is roughly 10 %lower than T1, corresponding to γ2 ' 650 s−1.The shift induced on the clock transition by the buffer gas

is +8555(5) Hz, consistent with a total buffer gas pressureof 49(2) torr, assuming negligible error on the buffer gascomposition [22].

A finer tuning of the clock operating parameters (opticalpower, beam waist, cycle time, etc...) is needed, seeking forthe ultimate stability performances. However, a stability below5× 10−13 at 1 s is compatible with the clock signal shownin Fig. 6, considering the major noise sources of the currentsetup (including laser RIN and frequency noise, detector noise,etc...). This estimate is confirmed by preliminary measure-ments and will be further characterized in future works.

IV. CONCLUSIONS

A compact cavity-cell assembly, based on high-grade purityalumina as loading material has been designed, realized andcharacterized in terms of the main parameters of interest formicrowave clock applications. The dielectric loading has leadto a reduction of a factor 10 of the inner cavity volume, com-pared to the traditional cylindrical cavity. The size reductionwas achieved by maintaining the favorable field uniformity ofthe TE011 mode and high cavity quality factor.

Despite the scaling of the clock cell dimension, introducedto push the miniaturization, we achieved high fringe contrastand high atomic-line quality factor by integrating the cavityin a POP clock setup. The obtained clock signal is compatiblewith a high-performing vapor-cell Rb clock, with short-termin the mid 10−13 τ−1/2, as shown by preliminary character-ization. The reduced size of the assembly facilitates thermaluniformity, with foreseen benefits in the medium-long termstability.

The loaded-cavity approach thus adds to the existing designoptions for the realization of compact vapor cell clocks based

6

on microwave interrogation. As shown in [35], more refinedconfigurations can also be studied to make the design evenmore robust against environmental sensitivities. In this paperwe reported a quality factor rather high for the POP withoptical detection. If needed, this value can be tailored by usingalumina with different percentages of impurities.

The proposed loaded-cavity assembly paves the way for astrongly miniaturized Rb clock physics package with low massand power consumption, particularly appealing for spaceborneapplications.

ACKNOWLEDGMENTS

The authors thank Elio K. Bertacco for precious technicalhelp and Marwan S.p.A. (Pisa, Italy) for the cell filling. Weacknowledge valuable input on the alumina production processfrom Jörg-Uwe Wichert from Wesgo Ceramics (Erlangen,Germany) and Wesgo Ceramics GmbH for the alumina piecesprocurement. We also thank the LED laboratory staff fromPolitecnico di Torino for providing the CST software andcomputational facility.

REFERENCES

[1] W. J. Riley, A History of the Rubidium Frequency Standard. IEEEUFFC-S History, 2019.

[2] A. Godone, F. Levi, C. E. Calosso, and S. Micalizio, “High-performingvapor-cell frequency standards,” Riv. Nuovo Cimento, vol. 38, pp. 133–171, Mar. 2015.

[3] J. C. Camparo and T. U. Driskell, “The mercury-ion clock and thepulsed-laser rubidium clock: Near-term candidates for future GPS de-ployment,” Tech. Rep. TOR-2015-03893, Aerospace Corporation, 2015.

[4] B. L. Schmittberger and D. R. Scherer, “A review of contemporaryatomic frequency standards.” arXiv:2004.09987, 2020.

[5] P. Arpesi, J. Belfi, M. Gioia, N. Marzoli, R. Romani, A. Sapia,M. Gozzelino, C. Calosso, F. Levi, S. Micalizio, A. Tuozzi, and M. Bel-loni, “Rubidium pulsed optically pumped clock for space industry,” in2019 Joint Conference of the IEEE International Frequency ControlSymposium and European Frequency and Time Forum (EFTF/IFC),IEEE, apr 2019.

[6] S. Micalizio, A. Godone, C. Calosso, F. Levi, C. Affolderbach, andF. Gruet, “Pulsed optically pumped rubidium clock with high frequency-stability performance,” IEEE Trans. Ultrason., Ferroelectr., Freq. Con-trol, vol. 59, pp. 457–462, mar 2012.

[7] T. Bandi, C. Affolderbach, C. Stefanucci, F. Merli, A. K. Skrivervik,and G. Mileti, “Compact high-performance continuous-wave double-resonance rubidium standard with 1.4 × 10−13 τ−1/2 stability,” IEEETrans. Ultrason., Ferroelectr., Freq. Control, vol. 61, pp. 1769–1778,November 2014.

[8] A. Hudson and J. Camparo, “Mesoscopic physics in vapor-phase atomicsystems: Collision-shift gradients and the 0-0 hyperfine transition,” Phys.Rev. A, vol. 98, oct 2018.

[9] S. Micalizio, C. E. Calosso, A. Godone, and F. Levi, “Metrological char-acterization of the pulsed rb clock with optical detection,” Metrologia,vol. 49, no. 4, p. 425, 2012.

[10] A. Godone, S. Micalizio, F. Levi, and C. Calosso, “Microwave cav-ities for vapor cell frequency standards,” Rev. Sci. Instrum., vol. 82,p. 074703, jul 2011.

[11] W. Froncisz and J. S. Hyde, “The loop-gap resonator: a new microwavelumped circuit ESR sample structure,” J. Magn. Reson., vol. 47, pp. 515–521, may 1982.

[12] T. Sphicopoulos and F. Gardiol, “Slotted tube cavity: a compact res-onator with empty core,” IEE Proceedings H - Microwaves, Antennasand Propagation, vol. 134, no. 5, p. 405, 1987.

[13] C. Stefanucci, T. Bandi, F. Merli, M. Pellaton, C. Affolderbach,G. Mileti, and A. K. Skrivervik, “Compact microwave cavity for highperformance rubidium frequency standards,” Rev. Sci. Instrum., vol. 83,p. 104706, oct 2012.

[14] S. Kang, M. Gharavipour, C. Affolderbach, F. Gruet, and G. Mileti,“Demonstration of a high-performance pulsed optically pumped rb clockbased on a compact magnetron-type microwave cavity,” J. Appl. Phys.,vol. 117, p. 104510, mar 2015.

[15] Q. Hao, W. Xue, W. Li, F. Xu, X. Wang, W. Guo, P. Yun, andS. Zhang, “Microwave pulse-coherent technique based clock with anovel magnetron-type cavity,” IEEE Trans. Ultrason., Ferroelectr., Freq.Control, vol. 67, no. 4, pp. 873 – 878, 2019.

[16] D. A. Howe and F. L. Walls, “A compact hydrogen maser withexceptional long-term stability,” IEEE Trans. Instrum. Meas., vol. 32,pp. 218–223, mar 1983.

[17] H. E. Williams, T. M. Kwon, and T. McClelland, “Compact rectangularcavity for rubidium vapor cell frequency standards,” in 37th AnnualSymposium on Frequency Control, pp. 12–17, June 1983.

[18] R. R. Mett, W. Froncisz, and J. S. Hyde, “Axially uniform resonantcavity modes for potential use in electron paramagnetic resonancespectroscopy,” Rev. Sci. Instrum., vol. 72, pp. 4188–4200, nov 2001.

[19] K. Wang, Z. Du, Z. Yu, Y. Tian, Y. Liu, and S. Zhang, “A novel kind ofmicrowave cavity with low temperature sensitivity and high uniformityfor POP rubidium frequency standards,” in 2019 Joint Conference ofthe IEEE International Frequency Control Symposium and EuropeanFrequency and Time Forum (EFTF/IFC), IEEE, apr 2019.

[20] C. E. Calosso, M. Gozzelino, A. Godone, H. Lin, F. Levi, and S. Mical-izio, “Intensity detection noise in pulsed vapor-cell frequency standards,”IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 67, no. 5,pp. 1074–1079, 2020.

[21] S. Micalizio, A. Godone, F. Levi, and C. Calosso, “Pulsed opticallypumped 87Rb vapor cell frequency standard: A multilevel approach,”Phys. Rev. A, vol. 79, p. 013403, Jan 2009.

[22] J. Vanier, R. Kunski, N. Cyr, J. Y. Savard, and M. Tłtu, “On hyperfinefrequency shifts caused by buffer gases: Application to the opticallypumped passive rubidium frequency standard,” J. Appl. Phys., vol. 53,no. 8, pp. 5387–5391, 1982.

[23] J. Vanier and C. Audoin, The Quantum Physics of Atomic FrequencyStandards. Adam Hilger, 1989.

[24] https://www.3ds.com/products-services/simulia/products/cst-studio-suite/.

[25] Morgan Advanced Materials, WESGO Ceramics GmbH, AL995TMdatasheet.

[26] V. Gerginov, N. Nemitz, and S. Weyers, “Initial atomic coherences andramsey frequency pulling in fountain clocks,” Phys. Rev. A, vol. 90, sep2014.

[27] A. Godone, S. Micalizio, and F. Levi, “Pulsed optically pumped fre-quency standard,” Phys. Rev. A, vol. 70, aug 2004.

[28] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade,Photonic crystals: Molding the flow of light. Princeton University Press,2nd ed., 2008.

[29] C. Calosso, S. Micalizio, A. Godone, E. Bertacco, and F. Levi, “Electron-ics for the pulsed rubidium clock: Design and characterization,” IEEETrans. Ultrason., Ferroelectr., Freq. Control, vol. 54, pp. 1731–1740,sep 2007.

[30] C. E. Calosso, M. Gozzelino, E. Bertacco, S. Micalizio, B. Franois, andP. Yun, “Generalized electronics for compact atomic clocks,” in 2017Joint Conference of the European Frequency and Time Forum and IEEEInternational Frequency Control Symposium (EFTF/IFCS), pp. 322–323,July 2017.

[31] B. François, C. E. Calosso, M. Abdel Hafiz, S. Micalizio, and R. Boudot,“Simple-design ultra-low phase noise microwave frequency synthesizersfor high-performing Cs and Rb vapor-cell atomic clocks,” Rev. Sci.Instrum., vol. 86, no. 9, p. 094707, 2015.

[32] W. Franzen, “Spin relaxation of optically aligned rubidium vapor,” Phys.Rev., vol. 115, pp. 850–856, aug 1959.

[33] M. Gharavipour, C. Affolderbach, F. Gruet, I. S. Radojičić, A. J. Krmpot,B. M. Jelenković, and G. Mileti, “Optically-detected spin-echo methodfor relaxation times measurements in a rb atomic vapor,” New J. Phys.,vol. 19, p. 063027, jun 2017.

[34] S. Micalizio, C. E. Calosso, F. Levi, and A. Godone, “Ramsey-fringeshape in an alkali-metal vapor cell with buffer gas,” Phys. Rev. A, vol. 88,sep 2013.

[35] N. R. Wang, R. F. Yang, T. Z. Zhou, and L. S. Gao, “Fre-quency–temperature compensated sapphire loaded cavity for compacthydrogen masers,” Metrologia, vol. 45, pp. 534–538, sep 2008.

https://www.3ds.com/products-services/simulia/products/cst-studio-suite/https://www.3ds.com/products-services/simulia/products/cst-studio-suite/

I IntroductionII Cylindrical loaded-cavity designII-A Finite element analysis (FEA)

III Experimental characterizationIV ConclusionsReferences